550 research outputs found
The Graph Curvature Calculator and the curvatures of cubic graphs
We classify all cubic graphs with either non-negative Ollivier-Ricci
curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both
curvature notions that the non-negatively curved graphs are the prism graphs
and the M\"obius ladders. We also highlight an online tool for calculating the
curvature of graphs under several variants of these curvature notions that we
use in the classification. As a consequence of the classification result we
show, that non-negatively curved cubic expanders do not exist
The spectrum on prism graph using circulant matrix
Spectral graph theory discusses about the algebraic properties of graphs based on the spectrum of a graph. This article investigated the spectrum of prism graph. The method used in this research is the circulant matrix. The results showed that prism graph P2,s is a regular graph of degree 3, for s odd and s ≥ 3, P2,s is a circulantt graph with regular spectrum
Symmetry and Complexity
Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry
Proceedings of the 2018 Joint Workshop of Fraunhofer IOSB and Institute for Anthropomatics, Vision and Fusion Laboratory
The Proceeding of the annual joint workshop of the Fraunhofer IOSB and the Vision and Fusion
Laboratory (IES) 2018 of the KIT contain technical reports of the PhD-stundents on the status of their
research. The discussed topics ranging from computer vision and optical
metrology to network security and machine learning.
This volume provides a comprehensive and up-to-date overview of the research program of the IES
Laboratory and the Fraunhofer IOSB
Structured networks and coarse-grained descriptions: a dynamical perspective
This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics. In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions. In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focusing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance. In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions
- …